 In this video, we'll sketch a couple of graphs in polar coordinates. Consider the equation of the polar curve r equals sine of theta. We'll plot some points to sketch out the curve. First, we'll create a table of r and theta values, and I'm going to pick theta values that give me nice kind of clean known values for r. So let's start with theta being zero, sine of zero is zero. We've got pi over six, sine of pi over six is one-half, pi over three, sine of that is root three over two, pi over two, sine of pi over two is one, and then we proceed with known values from the unit circle. I'll fill out this chart, and at two pi, I'm back at zero. So we have our table here. When theta is zero, r is zero. When theta is pi over six, so we're looking at counterclockwise direction, pi over six, we have a value of one-half. We'll say this is one. So that's one over six, I have a radius of a half. At pi over three, I have a radius of root three over two. At pi over two, I have a radius of one and so on. And back at pi, I'm at zero. At seven pi over six, at seven pi over six, I notice that my r value is negative a half. If it were a half, then my coordinate would be out here. But because the r value is negative one-half, we actually plot it 180 degrees across. So seven pi over six, comma, negative one-half is plotted at this point. Four pi over three, so there's my angle. Four pi over three, negative root three over two is plotted here. Three pi over two as an angle measure, comma, negative one for my r value is plotted up here. Five pi over three, comma, negative root three over two is plotted here. Eleven pi over six, comma, negative one-half is plotted here. And then we're back to two pi zero. So here's my curve. Let's consider a second polar curve, r equals cosine of two theta. In this case, instead of plotting points, we're going to employ a little bit of a different approach to sketching the graph. Now when theta increases from zero to pi over four, we know that r ranges from one to zero. Now the direction is counterclockwise as r is one when theta is zero, and it's zero when theta is pi over four. Again, r is one when theta is zero, and it's zero when theta is pi over four. So I start here, and the orientation is this direction. So that was when theta was ranging from zero to pi over four, r was ranging between one and zero in that order. When theta increases from pi over four to pi over two, we know that r ranges from zero to negative one. So we started from zero to pi over four. This increment now, we're looking from pi over four, these are theta values, to pi over two, r ranges between zero and negative one. When theta is between pi over two and three pi over four, r ranges from negative one to zero. When theta is between three pi over four to pi, r ranges from zero to one. Now this coordinate is at negative one zero because we're looking when theta is pi, so we're already oriented along this negative x-axis, and the value at pi of r is one. So our coordinate in polar coordinates is pi one. This is my theta, and this is my r value. It translates to negative one zero in rectangular coordinates. Let's proceed. When theta increases from pi to five pi over four, r ranges from one to zero. So when theta is pi over four, five pi over four, which would be in this direction, that's my theta being five pi over four, the value is zero for r. So we see we come back to the origin. I'm going to change colors here just to give you a little bit of a clearer picture here. Now let's take a look when theta is between five pi over four and three pi over two. Now you may be wondering why I'm choosing these theta values. They're essentially the theta values that give me points at which my r value is a zero or a one or negative one. And that allows me to at least connect the points knowing the orientation of the curve. And I've picked the points that will give me those nice clean values. You don't have to do it this way, but this is one approach. Between five pi over four and three pi over two, let's take a look at what r would be. We range between zero, so cosine of two times five pi over four would give me cosine of five pi over two, which is zero. So we range from zero when theta is three pi over two. I obtain cosine of two times that. That gives me negative one. So at three pi over two as my angle measure, the r value is negative one, which brings me up here. When theta increases from three pi over two to seven pi over four, r ranges from negative one. So three pi over two comma negative one gives me this coordinate to zero. So we come back. And when theta finally increases from seven pi over four to two pi, cosine of two times seven pi over four is the zero here. Two pi in place of theta gives me r being one. So I come back, connect. And this curve is actually called a four petal rose. Let's confirm this graph using our calculator. And here we have cosine of two theta in our calculator. And we'll graph it. And there's our four petal rose.